Shape theorem and surface fluctuation for Poisson cylinders

Abstract

We prove a shape theorem for Poisson cylinders, and give a power-law bound on surface fluctuations. In particular, we show that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, if a point belongs to this set and has Euclidean norm below $R$, then this point lies at internal distance less than $R + O(R^{a})$ from the origin.